351 research outputs found
The Phase Diagram of Crystalline Surfaces
We report the status of a high-statistics Monte Carlo simulation of
non-self-avoiding crystalline surfaces with extrinsic curvature on lattices of
size up to nodes. We impose free boundary conditions. The free energy
is a gaussian spring tethering potential together with a normal-normal bending
energy. Particular emphasis is given to the behavior of the model in the cold
phase where we measure the decay of the normal-normal correlation function.Comment: 9 pages latex (epsf), 4 EPS figures, uuencoded and compressed.
Contribution to Lattice '9
New Renormalization Group Results for Scaling of Self-Avoiding Tethered Membranes
The scaling properties of self-avoiding polymerized 2-dimensional membranes
are studied via renormalization group methods based on a multilocal operator
product expansion. The renormalization group functions are calculated to second
order. This yields the scaling exponent nu to order epsilon^2. Our
extrapolations for nu agree with the Gaussian variational estimate for large
space dimension d and are close to the Flory estimate for d=3. The interplay
between self-avoidance and rigidity at small d is briefly discussed.Comment: 97 pages, 120 .eps-file
Quaternion Singular Value Decomposition based on Bidiagonalization to a Real Matrix using Quaternion Householder Transformations
We present a practical and efficient means to compute the singular value
decomposition (svd) of a quaternion matrix A based on bidiagonalization of A to
a real bidiagonal matrix B using quaternionic Householder transformations.
Computation of the svd of B using an existing subroutine library such as lapack
provides the singular values of A. The singular vectors of A are obtained
trivially from the product of the Householder transformations and the real
singular vectors of B. We show in the paper that left and right quaternionic
Householder transformations are different because of the noncommutative
multiplication of quaternions and we present formulae for computing the
Householder vector and matrix in each case
Theta-point universality of polyampholytes with screened interactions
By an efficient algorithm we evaluate exactly the disorder-averaged
statistics of globally neutral self-avoiding chains with quenched random charge
in monomer i and nearest neighbor interactions on
square (22 monomers) and cubic (16 monomers) lattices. At the theta transition
in 2D, radius of gyration, entropic and crossover exponents are well compatible
with the universality class of the corresponding transition of homopolymers.
Further strong indication of such class comes from direct comparison with the
corresponding annealed problem. In 3D classical exponents are recovered. The
percentage of charge sequences leading to folding in a unique ground state
approaches zero exponentially with the chain length.Comment: 15 REVTEX pages. 4 eps-figures . 1 tabl
First-order transition of tethered membranes in 3d space
We study a model of phantom tethered membranes, embedded in three-dimensional
space, by extensive Monte Carlo simulations. The membranes have hexagonal
lattice structure where each monomer is interacting with six nearest-neighbors
(NN). Tethering interaction between NN, as well as curvature penalty between NN
triangles are taken into account. This model is new in the sense that NN
interactions are taken into account by a truncated Lennard-Jones potential
including both repulsive and attractive parts. The main result of our study is
that the system undergoes a first-order crumpling transition from low
temperature flat phase to high temperature crumpled phase, in contrast with
early numerical results on models of tethered membranes.Comment: 5 pages, 6 figure
M.C.R.G. Study of Fixed-connectivity Surfaces
We apply Monte Carlo Renormalization group to the crumpling transition in
random surface models of fixed connectivity. This transition is notoriously
difficult to treat numerically. We employ here a Fourier accelerated Langevin
algorithm in conjunction with a novel blocking procedure in momentum space
which has proven extremely successful in . We perform two
successive renormalizations in lattices with up to sites. We obtain a
result for the critical exponent in general agreement with previous
estimates and similar error bars, but with much less computational effort. We
also measure with great accuracy . As a by-product we are able to
determine the fractal dimension of random surfaces at the crumpling
transition.Comment: 35 pages,Latex file, 6 Postscript figures uuencoded,uses psfig.sty 2
misspelled references corrected and one added. Paper unchange
The Statistical Mechanics of Membranes
The fluctuations of two-dimensional extended objects membranes is a rich and
exciting field with many solid results and a wide range of open issues. We
review the distinct universality classes of membranes, determined by the local
order, and the associated phase diagrams. After a discussion of several
physical examples of membranes we turn to the physics of crystalline (or
polymerized) membranes in which the individual monomers are rigidly bound. We
discuss the phase diagram with particular attention to the dependence on the
degree of self-avoidance and anisotropy. In each case we review and discuss
analytic, numerical and experimental predictions of critical exponents and
other key observables. Particular emphasis is given to the results obtained
from the renormalization group epsilon-expansion. The resulting renormalization
group flows and fixed points are illustrated graphically. The full technical
details necessary to perform actual calculations are presented in the
Appendices. We then turn to a discussion of the role of topological defects
whose liberation leads to the hexatic and fluid universality classes. We finish
with conclusions and a discussion of promising open directions for the future.Comment: 75 LaTeX pages, 36 figures. To appear in Physics Reports in the
Proceedings of RG2000, Taxco, 199
Universal Negative Poisson Ratio of Self Avoiding Fixed Connectivity Membranes
We determine the Poisson ratio of self-avoiding fixed-connectivity membranes,
modeled as impenetrable plaquettes, to be sigma=-0.37(6), in statistical
agreement with the Poisson ratio of phantom fixed-connectivity membranes
sigma=-0.32(4). Together with the equality of critical exponents, this result
implies a unique universality class for fixed-connectivity membranes. Our
findings thus establish that physical fixed-connectivity membranes provide a
wide class of auxetic (negative Poisson ratio) materials with significant
potential applications in materials science.Comment: 4 pages, 3 figures, LaTeX (revtex) Published version - title changed,
one figure improved and one reference change
Fundamental representations and algebraic properties of biquaternions or complexified quaternions
The fundamental properties of biquaternions (complexified quaternions) are
presented including several different representations, some of them new, and
definitions of fundamental operations such as the scalar and vector parts,
conjugates, semi-norms, polar forms, and inner and outer products. The notation
is consistent throughout, even between representations, providing a clear
account of the many ways in which the component parts of a biquaternion may be
manipulated algebraically
Generalizing the O(N)-field theory to N-colored manifolds of arbitrary internal dimension D
We introduce a geometric generalization of the O(N)-field theory that
describes N-colored membranes with arbitrary dimension D. As the O(N)-model
reduces in the limit N->0 to self-avoiding polymers, the N-colored manifold
model leads to self-avoiding tethered membranes. In the other limit, for inner
dimension D->1, the manifold model reduces to the O(N)-field theory. We analyze
the scaling properties of the model at criticality by a one-loop perturbative
renormalization group analysis around an upper critical line. The freedom to
optimize with respect to the expansion point on this line allows us to obtain
the exponent \nu of standard field theory to much better precision that the
usual 1-loop calculations. Some other field theoretical techniques, such as the
large N limit and Hartree approximation, can also be applied to this model. By
comparison of low and high temperature expansions, we arrive at a conjecture
for the nature of droplets dominating the 3d-Ising model at criticality, which
is satisfied by our numerical results. We can also construct an appropriate
generalization that describes cubic anisotropy, by adding an interaction
between manifolds of the same color. The two parameter space includes a variety
of new phases and fixed points, some with Ising criticality, enabling us to
extract a remarkably precise value of 0.6315 for the exponent \nu in d=3. A
particular limit of the model with cubic anisotropy corresponds to the random
bond Ising problem; unlike the field theory formulation, we find a fixed point
describing this system at 1-loop order.Comment: 57 pages latex, 26 figures included in the tex
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